reference help needed on a fact about decomposition of representations

Question

For simplicity, let $G_n$ be $GL_n(\mathbb{R})$, $\mathfrak{g}_n$ be its Lie algebra. $K_n$ be $O(n)$. I want to know any reference about the following statement.

For any irreducible admissible $(\mathfrak{g}_n\oplus \mathfrak{g}_m,K_n\times K_m)$ module $V$, there exist an irreducible admissible $(\mathfrak{g}_n, K_n)$ module $U$, and an irreducible admissible $(\mathfrak{g}_m, K_m)$ module $W$, such that $V$ is isomorphic to $U\otimes W$. Both $U$ and $W$ are uniquely determined by $V$ up to isomorphism.

Similar of this is true for representations of finite groups, smooth representations of p-adic groups. It seems to me that this is also true for real reductive groups, and I can't find it in any standard reference. I appreciate a lot if anyone would provide some related paper or book.

Answer 1

Well it's slightly later than you probably would've liked, but here's a reference for exactly what you're looking for:

Gourevitch, D. and Kemarsky, A., 2012. Irreducible representations of product of real reductive groups. (arXiv link: https://arxiv.org/abs/1212.6004)

See Theorem 1.1 and its proof. There the authors show that each irreducible Harish-Chandra $(\mathfrak{g}_1 \times \mathfrak{g}_2, K_1 \times K_2)$-module has a unique decomposition into a tensor product of irreducible Harish-Chandra $(\mathfrak{g}_i, K_i)$-modules. (Here by a Harish-Chandra $(\mathfrak{g},K)$-module we mean a $(\mathfrak{g},K)$-module that is admissible and finitely generated as a $U(\mathfrak{g}_\mathbb{C})$-module.)

They also reference Aizenbud and the first author's 2009 paper wherein a converse to this is stated and proved in Appendix A. (That direction is easier, and IIRC their proof is maybe a few paragraphs in length.)